Total order

A total order on a set $X$ is a partial order $\le$ on $X$ such that for all $a,b\in X$, either $a\le b$ or $b\le a$.

Total orders are also called linear orders. A well-ordered set is a totally ordered set with the additional property that every nonempty subset has a least element.

Examples:

• The usual order $\le$ on $\mathbb{Z}$ (the integers ) is a total order.
• (Lexicographic order) If $X$ and $Y$ are totally ordered, define an order on $X\times Y$ by $(x,y)\le_{\mathrm{lex}}(x',y')$ if either $x<x'$, or $x=x'$ and $y\le y'$ (where $x<x'$ means $x\le x'$ and $x\ne x'$). This gives a total order on the Cartesian product $X\times Y$.